Solving Linear Equations and Systems

Single Variable Linear Equations

General form: ax + b = c

Solving: x = (c - b) / a

Example: 3x + 7 = 22 → 3x = 15 → x = 5

Two-Variable Systems

Substitution Method

1. Solve one equation for one variable
2. Substitute into the other equation
3. Solve, then back-substitute

Elimination Method

1. Multiply equations to match coefficients
2. Add/subtract to eliminate one variable
3. Solve for remaining variable

Example:

2x + 3y = 12  ...(1)
4x -  y =  2  ...(2)

Multiply (2) by 3: 12x - 3y = 6  ...(3)
Add (1) + (3): 14x = 18 → x = 9/7
Substitute back: y = (12 - 2×9/7)/3 = (12 - 18/7)/3 = 22/7

Three-Variable Systems

3 equations, 3 unknowns. Use elimination to reduce to 2×2, then solve.

Matrix Method (Cramer's Rule / Gaussian Elimination)

System [A]{x} = {b}

Gaussian Elimination steps:

1. Write augmented matrix [A|b]
2. Use row operations to get upper triangular form
3. Back-substitute to find unknowns

Cramer's Rule (2×2): For ax + by = e, cx + dy = f:

• D = ad - bc
• x = (ed - bf)/D
• y = (af - ec)/D

Engineering Application

Systems of equations appear everywhere in engineering:

• Truss analysis (method of joints)
• Electrical circuits (Kirchhoff's laws)
• Equilibrium of structures
• Fluid network balancing

Summary

Mastering linear systems — substitution, elimination, matrices — is non-negotiable for engineering. Practice until you can solve 3×3 systems efficiently.